↑

Properties

Label 2-21e2-21.5-c3-0-5
Degree $2$
Conductor $441$
Sign $-0.932 - 0.360i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.44 − 1.41i)2-s + (−10.5 + 18.2i)5-s + 22.6i·8-s + (51.6 − 29.7i)10-s + (13.4 − 7.77i)11-s + 29.7i·13-s + (32.0 − 55.4i)16-s + (31.6 + 54.7i)17-s + (77.4 + 44.6i)19-s − 44·22-s + (67.3 + 38.8i)23-s + (−159.5 − 276. i)25-s + (42.1 − 72.9i)26-s + 125. i·29-s + (−206. + 119. i)31-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)2-s + (−0.942 + 1.63i)5-s + 0.999i·8-s + (1.63 − 0.942i)10-s + (0.369 − 0.213i)11-s + 0.635i·13-s + (0.500 − 0.866i)16-s + (0.450 + 0.781i)17-s + (0.934 + 0.539i)19-s − 0.426·22-s + (0.610 + 0.352i)23-s + (−1.27 − 2.21i)25-s + (0.317 − 0.550i)26-s + 0.805i·29-s + (−1.19 + 0.690i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.932 - 0.360i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.932 - 0.360i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4188199107\)
\(L(\frac12)\) \(\approx\) \(0.4188199107\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (2.44 + 1.41i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (10.5 - 18.2i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-13.4 + 7.77i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 29.7iT - 2.19e3T^{2} \)
17 \( 1 + (-31.6 - 54.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-77.4 - 44.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-67.3 - 38.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 125. iT - 2.43e4T^{2} \)
31 \( 1 + (206. - 119. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-92 + 159. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 105.T + 6.89e4T^{2} \)
43 \( 1 + 190T + 7.95e4T^{2} \)
47 \( 1 + (21.0 - 36.4i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (309. - 178. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (42.1 + 72.9i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-567. - 327. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (148 + 256. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 329. iT - 3.57e5T^{2} \)
73 \( 1 + (-696. + 402. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (418 - 723. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.22e3T + 5.71e5T^{2} \)
89 \( 1 + (347. - 602. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 566. iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.05322795337324027258765672757, −10.31570630378083768776002354772, −9.473317149819231361013323027257, −8.428115649042219033752814590497, −7.53362807175057844252984693788, −6.73841236149433294976740107336, −5.50752923130097131578502486298, −3.86738728043747428983383428824, −2.97412603454199077630697671152, −1.52059210400712800111388328782, 0.23175003239712192148008083965, 1.08624955315239203648286585325, 3.42325399233126639062116539335, 4.50828262464631382098575311852, 5.42331770171821647606432001880, 7.00228093675756138535765513229, 7.80573898732337470597914721755, 8.396826400624938831909320017780, 9.277307320716043714857331031819, 9.774986071012583388330923059573

Graph of the $Z$-function along the critical line